is the Euclidean azimuth-rapidity distance of particle \(i\)  from the jet axis.
Our study is limited to a subset of the above defined general angularities,
namely the ones that exhibit infrared and collinear (IRC) safety. This poses
the restrictions \(\kappa =1\)  and \(\alpha >0\) .
Furthermore, it is well-known [1]}, [2]}
that IRC safe angularities with \(\alpha \le 1\)  are sensitive to recoil
against soft emissions, leading to a rather complicated resummation structure. To circumvent these additional complications,
we compute for these angularities the distance measure in Eq. (REF ) with respect to the jet axis
obtained by reclustering the jet constitutents with the anti-\(k_t\)  algorithm but using the Winner-Take-All (WTA) recombination
scheme [3]}.
We are also interested in computing angularities on groomed jets. In this case,
we recluster the jet with the Cambridge–Aachen (C/A) algorithm and consider
the SoftDrop grooming algorithm with parameters \(z_{\rm cut}\)  and \(\beta \)  [4]}. The angularity is then
computed on the resulting SoftDrop jet, i.e. both sums in Eq. (REF ) are restricted to the particles
that survived the grooming.
For groomed jets we also adopt the WTA prescription for angularities with \(\alpha \le 1\) .
